introduction-probability.pdf

But now we get e g n ? d p ϕ ? p ϕ b p ϕ 1 b ω 1i

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and can take sums and the equality remains true). But now we get E g n ( η ) d P ϕ ( η ) = P ϕ ( B ) = P ( ϕ - 1 ( B )) = Ω 1I ϕ - 1 ( B ) ( ω ) d P ( ω ) = Ω 1I B ( ϕ ( ω )) d P ( ω ) = Ω g n ( ϕ ( ω )) d P ( ω ) . Let us give an example for the change of variable formula. 1 In other words, P ϕ is the law of ϕ .
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3.5. FUBINI’S THEOREM 59 Definition 3.4.2 [Moments] Assume that n ∈ { 1 , 2 , .... } . (1) For a random variable f : Ω R the expected value E | f | n is called n -th absolute moment of f . If E f n exists, then E f n is called n -th moment of f . (2) For a probability measure μ on ( R , B ( R )) the expected value R | x | n ( x ) is called n -th absolute moment of μ . If R x n ( x ) exists, then R x n ( x ) is called n -th moment of μ . Corollary 3.4.3 Let , F , P ) be a probability space and f : Ω R be a random variable with law P f . Then, for all n = 1 , 2 , ... , E | f | n = R | x | n d P f ( x ) and E f n = R x n d P f ( x ) , where the latter equality has to be understood as follows: if one side exists, then the other exists as well and they coincide. If the law P f has a density p in the sense of Proposition 3.3.2, then R | x | n d P f ( x ) can be replaced by R | x | n p ( x ) dx and R x n d P f ( x ) by R x n p ( x ) dx . 3.5 Fubini’s Theorem In this section we consider iterated integrals, as they appear very often in applications, and show in Fubini ’s 2 Theorem that integrals with respect to product measures can be written as iterated integrals and that one can change the order of integration in these iterated integrals. In many cases this provides an appropriate tool for the computation of integrals. Before we start with Fubini ’s Theorem we need some preparations. First we recall the notion of a vector space. Definition 3.5.1 [vector space] A set L equipped with operations ”+” : L × L L and ” · ” : R × L L is called vector space over R if the following conditions are satisfied: (1) x + y = y + x for all x, y L . (2) x + ( y + z ) = ( x + y ) + z for all x, y, z L . (3) There exists a 0 L such that x + 0 = x for all x L . (4) For all x L there exists a - x such that x + ( - x ) = 0. 2 Guido Fubini, 19/01/1879 (Venice, Italy) - 06/06/1943 (New York, USA).
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60 CHAPTER 3. INTEGRATION (5) 1 x = x . (6) α ( βx ) = ( αβ ) x for all α, β R and x L . (7) ( α + β ) x = αx + βx for all α, β R and x L . (8) α ( x + y ) = αx + αy for all α R and x, y L . Usually one uses the notation x - y := x + ( - y ) and - x + y := ( - x ) + y etc. Now we state the Monotone Class Theorem . It is a powerful tool by which, for example, measurability assertions can be proved. Proposition 3.5.2 [Monotone Class Theorem] Let H be a class of bounded functions from Ω into R satisfying the following conditions: (1) H is a vector space over R where the natural point-wise operations ”+” and · are used. (2) 1I Ω H . (3) If f n H , f n 0 , and f n f , where f is bounded on Ω , then f H . Then one has the following: if H contains the indicator function of every set from some π -system I of subsets of Ω , then H contains every bounded σ ( I ) -measurable function on Ω .
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